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P02 · 装箱与覆盖 · 经典问题 · 基线易突破 · 难

单位圆内的等圆装箱 · n = 99

文献中也称circles in a circlepacking equal circles in a circlecci

在半径 1 的圆内放置 n 个半径相同、互不重叠的小圆，使共同半径尽可能大。

子题n = 99
目标最大化 共同半径

纪录对比

共同半径 · 越大越好
纪录数值 / 区间作者 / 持有人来源
外部已知最好[0.0906360198128194295912438099535, 0.0906360198128194295912438099545]见来源署名packomania.com ↗
本站纪录暂无选手纪录——

严格定义

  • 容器半径 1 的圆，圆心在 (1, 1)，两个坐标都在 0 到 2 之间
  • 提交恰好 n 个圆：一个共同半径 radius 与 n 个圆心 centers
  • 约束每个圆完整落在容器内；两两内部不重叠，相切允许
  • 目标让共同半径尽可能大
挑战这个纪录 ↗

当前展示：起步布局

2y0
0x2
已验证构造r = 0.090636019

帮助理解

哪里有优化空间

最优构形是「卡死」的接触结构：圆彼此顶住、顶住边界，常出现斜排、错位、以及不碰任何邻居的游离圆。规整的网格摆法几乎从不最优。

前沿在哪里

n = 1..14 与 19 已证明；其余规模的最优性仍未知。本站接入 Packomania 中 n = 1..300 的公开构型，大规模部分集中成一张表供算法批量挑战。

查看来源
单位圆内的等圆装箱 n = 99 的当前构型
构型与历史

起步布局

查看构型、求解笔记与纪录历史。

挑战这个纪录 ↗提交证明 / 思路 ↓在讨论区分享证明或思路，审核采纳后可获得证明分。
ANSWER FORMAT

答案怎么写

容器是半径 1 的圆，圆心在 (1, 1)，所以两个坐标都在 0 到 2 之间。坐标和半径用同一个单位，直接写成小数，例如 "0.5"，最多九位小数。

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.2"。

起步布局答案

{
  "centers": [
    [
      "0.90936398",
      "0.095164125"
    ],
    [
      "1.09063602",
      "0.095164125"
    ],
    [
      "1.26830653",
      "0.131118849"
    ],
    [
      "0.693762172",
      "0.143751647"
    ],
    [
      "1.435315563",
      "0.201599593"
    ],
    [
      "0.497669295",
      "0.248015447"
    ],
    [
      "0.833153661",
      "0.259637648"
    ],
    [
      "1.148333571",
      "0.267008662"
    ],
    [
      "1.585026817",
      "0.303805722"
    ],
    [
      "0.663097697",
      "0.322411218"
    ],
    [
      "1.315342603",
      "0.337489407"
    ],
    [
      "0.988650932",
      "0.352804976"
    ],
    [
      "0.349861848",
      "0.364183021"
    ],
    [
      "0.518224336",
      "0.431366538"
    ],
    [
      "1.711491338",
      "0.433675955"
    ],
    [
      "0.830217024",
      "0.440885899"
    ],
    [
      "1.144632383",
      "0.448242913"
    ],
    [
      "1.480298978",
      "0.452187288"
    ],
    [
      "0.236666607",
      "0.505768294"
    ],
    [
      "1.311641415",
      "0.518723657"
    ],
    [
      "0.671408631",
      "0.528289827"
    ],
    [
      "0.985315243",
      "0.534716037"
    ],
    [
      "1.809683909",
      "0.586049743"
    ],
    [
      "1.628628052",
      "0.59490011"
    ],
    [
      "0.510866183",
      "0.613270477"
    ],
    [
      "0.82650685",
      "0.622119966"
    ],
    [
      "1.139861735",
      "0.629452166"
    ],
    [
      "1.451605018",
      "0.633918025"
    ],
    [
      "0.334859178",
      "0.658142082"
    ],
    [
      "0.153803321",
      "0.666992449"
    ],
    [
      "0.666637984",
      "0.70949908"
    ],
    [
      "0.981053342",
      "0.716856094"
    ],
    [
      "1.294408228",
      "0.724188294"
    ],
    [
      "1.875702733",
      "0.754872334"
    ],
    [
      "1.57800708",
      "0.769655747"
    ],
    [
      "0.457161201",
      "0.79193957"
    ],
    [
      "0.821736202",
      "0.803329218"
    ],
    [
      "1.135599835",
      "0.811592222"
    ],
    [
      "0.104564661",
      "0.841449054"
    ],
    [
      "0.286332613",
      "0.852953588"
    ],
    [
      "1.418262185",
      "0.856550485"
    ],
    [
      "1.733338856",
      "0.867086767"
    ],
    [
      "0.613935942",
      "0.882940849"
    ],
    [
      "0.976282695",
      "0.898065347"
    ],
    [
      "1.903452241",
      "0.935978601"
    ],
    [
      "1.259453792",
      "0.943954413"
    ],
    [
      "1.572808677",
      "0.951286613"
    ],
    [
      "0.427381753",
      "0.970748789"
    ],
    [
      "0.769034161",
      "0.976770987"
    ],
    [
      "0.090907184",
      "1.022205867"
    ],
    [
      "1.100136652",
      "1.030427538"
    ],
    [
      "1.414000284",
      "1.038690542"
    ],
    [
      "1.75526025",
      "1.047028437"
    ],
    [
      "0.268577694",
      "1.058160591"
    ],
    [
      "0.609165295",
      "1.064150101"
    ],
    [
      "0.923580653",
      "1.071507115"
    ],
    [
      "1.254683144",
      "1.125163666"
    ],
    [
      "1.569098503",
      "1.13252068"
    ],
    [
      "0.449848155",
      "1.150623226"
    ],
    [
      "0.764263513",
      "1.15798024"
    ],
    [
      "1.894651739",
      "1.162914438"
    ],
    [
      "0.113373585",
      "1.202080305"
    ],
    [
      "1.095366004",
      "1.211636791"
    ],
    [
      "1.41029011",
      "1.219924608"
    ],
    [
      "1.724595774",
      "1.225688008"
    ],
    [
      "0.291044096",
      "1.238035028"
    ],
    [
      "0.604394647",
      "1.245359354"
    ],
    [
      "0.918810006",
      "1.252716368"
    ],
    [
      "1.25097297",
      "1.306397733"
    ],
    [
      "1.566161866",
      "1.313768931"
    ],
    [
      "0.445187202",
      "1.333426093"
    ],
    [
      "0.759492866",
      "1.339189493"
    ],
    [
      "0.171071136",
      "1.373924842"
    ],
    [
      "1.828928864",
      "1.373924842"
    ],
    [
      "1.09165583",
      "1.392870857"
    ],
    [
      "1.405519463",
      "1.401133861"
    ],
    [
      "0.600684473",
      "1.426593421"
    ],
    [
      "0.915099832",
      "1.433950435"
    ],
    [
      "1.560622334",
      "1.494956309"
    ],
    [
      "1.260976831",
      "1.510527554"
    ],
    [
      "0.442250564",
      "1.514674344"
    ],
    [
      "0.755782692",
      "1.520423559"
    ],
    [
      "0.261707156",
      "1.530911033"
    ],
    [
      "1.738292844",
      "1.530911033"
    ],
    [
      "1.091233781",
      "1.574142406"
    ],
    [
      "1.416079702",
      "1.604350002"
    ],
    [
      "0.910948485",
      "1.615699814"
    ],
    [
      "0.581642053",
      "1.630560344"
    ],
    [
      "1.618319885",
      "1.666800846"
    ],
    [
      "1.227896396",
      "1.693241386"
    ],
    [
      "0.411586089",
      "1.693333915"
    ],
    [
      "0.748436809",
      "1.701546695"
    ],
    [
      "1.03651277",
      "1.746957808"
    ],
    [
      "1.473777253",
      "1.776194539"
    ],
    [
      "0.605013449",
      "1.812846539"
    ],
    [
      "1.310408522",
      "1.854745224"
    ],
    [
      "0.77443542",
      "1.880944646"
    ],
    [
      "1.134705341",
      "1.899331596"
    ],
    [
      "0.953649484",
      "1.908181963"
    ]
  ],
  "radius": "0.090636019"
}
提交格式与技术细节需要编写程序或准备 JSON 答案时再查看＋

子题参数

{
  "n": 99
}

起步布局答案

{
  "centers": [
    [
      "0.90936398",
      "0.095164125"
    ],
    [
      "1.09063602",
      "0.095164125"
    ],
    [
      "1.26830653",
      "0.131118849"
    ],
    [
      "0.693762172",
      "0.143751647"
    ],
    [
      "1.435315563",
      "0.201599593"
    ],
    [
      "0.497669295",
      "0.248015447"
    ],
    [
      "0.833153661",
      "0.259637648"
    ],
    [
      "1.148333571",
      "0.267008662"
    ],
    [
      "1.585026817",
      "0.303805722"
    ],
    [
      "0.663097697",
      "0.322411218"
    ],
    [
      "1.315342603",
      "0.337489407"
    ],
    [
      "0.988650932",
      "0.352804976"
    ],
    [
      "0.349861848",
      "0.364183021"
    ],
    [
      "0.518224336",
      "0.431366538"
    ],
    [
      "1.711491338",
      "0.433675955"
    ],
    [
      "0.830217024",
      "0.440885899"
    ],
    [
      "1.144632383",
      "0.448242913"
    ],
    [
      "1.480298978",
      "0.452187288"
    ],
    [
      "0.236666607",
      "0.505768294"
    ],
    [
      "1.311641415",
      "0.518723657"
    ],
    [
      "0.671408631",
      "0.528289827"
    ],
    [
      "0.985315243",
      "0.534716037"
    ],
    [
      "1.809683909",
      "0.586049743"
    ],
    [
      "1.628628052",
      "0.59490011"
    ],
    [
      "0.510866183",
      "0.613270477"
    ],
    [
      "0.82650685",
      "0.622119966"
    ],
    [
      "1.139861735",
      "0.629452166"
    ],
    [
      "1.451605018",
      "0.633918025"
    ],
    [
      "0.334859178",
      "0.658142082"
    ],
    [
      "0.153803321",
      "0.666992449"
    ],
    [
      "0.666637984",
      "0.70949908"
    ],
    [
      "0.981053342",
      "0.716856094"
    ],
    [
      "1.294408228",
      "0.724188294"
    ],
    [
      "1.875702733",
      "0.754872334"
    ],
    [
      "1.57800708",
      "0.769655747"
    ],
    [
      "0.457161201",
      "0.79193957"
    ],
    [
      "0.821736202",
      "0.803329218"
    ],
    [
      "1.135599835",
      "0.811592222"
    ],
    [
      "0.104564661",
      "0.841449054"
    ],
    [
      "0.286332613",
      "0.852953588"
    ],
    [
      "1.418262185",
      "0.856550485"
    ],
    [
      "1.733338856",
      "0.867086767"
    ],
    [
      "0.613935942",
      "0.882940849"
    ],
    [
      "0.976282695",
      "0.898065347"
    ],
    [
      "1.903452241",
      "0.935978601"
    ],
    [
      "1.259453792",
      "0.943954413"
    ],
    [
      "1.572808677",
      "0.951286613"
    ],
    [
      "0.427381753",
      "0.970748789"
    ],
    [
      "0.769034161",
      "0.976770987"
    ],
    [
      "0.090907184",
      "1.022205867"
    ],
    [
      "1.100136652",
      "1.030427538"
    ],
    [
      "1.414000284",
      "1.038690542"
    ],
    [
      "1.75526025",
      "1.047028437"
    ],
    [
      "0.268577694",
      "1.058160591"
    ],
    [
      "0.609165295",
      "1.064150101"
    ],
    [
      "0.923580653",
      "1.071507115"
    ],
    [
      "1.254683144",
      "1.125163666"
    ],
    [
      "1.569098503",
      "1.13252068"
    ],
    [
      "0.449848155",
      "1.150623226"
    ],
    [
      "0.764263513",
      "1.15798024"
    ],
    [
      "1.894651739",
      "1.162914438"
    ],
    [
      "0.113373585",
      "1.202080305"
    ],
    [
      "1.095366004",
      "1.211636791"
    ],
    [
      "1.41029011",
      "1.219924608"
    ],
    [
      "1.724595774",
      "1.225688008"
    ],
    [
      "0.291044096",
      "1.238035028"
    ],
    [
      "0.604394647",
      "1.245359354"
    ],
    [
      "0.918810006",
      "1.252716368"
    ],
    [
      "1.25097297",
      "1.306397733"
    ],
    [
      "1.566161866",
      "1.313768931"
    ],
    [
      "0.445187202",
      "1.333426093"
    ],
    [
      "0.759492866",
      "1.339189493"
    ],
    [
      "0.171071136",
      "1.373924842"
    ],
    [
      "1.828928864",
      "1.373924842"
    ],
    [
      "1.09165583",
      "1.392870857"
    ],
    [
      "1.405519463",
      "1.401133861"
    ],
    [
      "0.600684473",
      "1.426593421"
    ],
    [
      "0.915099832",
      "1.433950435"
    ],
    [
      "1.560622334",
      "1.494956309"
    ],
    [
      "1.260976831",
      "1.510527554"
    ],
    [
      "0.442250564",
      "1.514674344"
    ],
    [
      "0.755782692",
      "1.520423559"
    ],
    [
      "0.261707156",
      "1.530911033"
    ],
    [
      "1.738292844",
      "1.530911033"
    ],
    [
      "1.091233781",
      "1.574142406"
    ],
    [
      "1.416079702",
      "1.604350002"
    ],
    [
      "0.910948485",
      "1.615699814"
    ],
    [
      "0.581642053",
      "1.630560344"
    ],
    [
      "1.618319885",
      "1.666800846"
    ],
    [
      "1.227896396",
      "1.693241386"
    ],
    [
      "0.411586089",
      "1.693333915"
    ],
    [
      "0.748436809",
      "1.701546695"
    ],
    [
      "1.03651277",
      "1.746957808"
    ],
    [
      "1.473777253",
      "1.776194539"
    ],
    [
      "0.605013449",
      "1.812846539"
    ],
    [
      "1.310408522",
      "1.854745224"
    ],
    [
      "0.77443542",
      "1.880944646"
    ],
    [
      "1.134705341",
      "1.899331596"
    ],
    [
      "0.953649484",
      "1.908181963"
    ]
  ],
  "radius": "0.090636019"
}

提交 radius 与 centers。每个数写成十进制字符串，例如 "0.2"。 · 验证器 v1.0.0

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